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Katsanikas, M., & Wiggins, S. (2018). Phase Space Structure And Katsanikas, M., & Wiggins, S. (2018). Phase space structure and transport in a caldera potential energy surface. International Journal of Bifurcation and Chaos, 28(13), [1830042]. https://doi.org/10.1142/S0218127418300422 Peer reviewed version License (if available): Other Link to published version (if available): 10.1142/S0218127418300422 Link to publication record in Explore Bristol Research PDF-document This is the accepted author manuscript (AAM). The final published version (version of record) is available online via Wiley at https://doi.org/10.1142/S0218127418300422 . Please refer to any applicable terms of use of the publisher. University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ October 11, 2018 0:49 katsanikasandwiggins International Journal of Bifurcation and Chaos c World Scientific Publishing Company Phase Space Structure and Transport in a Caldera Potential Energy Surface MATTHAIOS KATSANIKAS AND STEPHEN WIGGINS School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK [email protected], [email protected] Received (to be inserted by publisher) We study phase space transport in a 2D caldera potential energy surface (PES) using techniques from nonlinear dynamics. The caldera PES is characterized by a flat region or shallow minimum at its center surrounded by potential walls and multiple symmetry related index one saddle points that allow entrance and exit from this intermediate region. We have discovered four qualitatively distinct cases of the structure of the phase space that govern phase space transport. These cases are categorized according to the total energy and the stability of the periodic orbits associated with the family of the central minimum, the bifurcations of the same family, and the energetic accessibility of the index one saddles. In each case we have computed the invariant manifolds of the unstable periodic orbits of the central region of the potential and the invariant manifolds of the unstable periodic orbits of the families of periodic orbits associated with the index one saddles. The periodic orbits of the central region are, for the first case, the unstable periodic orbits with period 10 that are outside the stable region of the stable periodic orbits of the family of the central minimum. In addition, the periodic orbits of the central region are, for the second and third case, the unstable periodic orbits of the family of the central minimum and for the fourth case the unstable periodic orbits with period 2 of a period-doubling bifurcation of the family of the central minimum. We have found that there are three distinct mechanisms determined by the invariant manifold structure of the unstable periodic orbits govern the phase space transport. The first mechanism explains the nature of the entrance of the trajectories from the region of the low energy saddles into the caldera and how they may become trapped in the central region of the potential. The second mechanism describes the trapping of the trajectories that begin from the central region of the caldera, their transport to the regions of the saddles, and the nature of their exit from the caldera. The third mechanism describes the arXiv:1809.06702v2 [nlin.CD] 9 Oct 2018 phase space geometry responsible for the dynamical matching of trajectories originally proposed by Carpenter and described in [Collins et al., 2014] for the two dimensional caldera PES that we consider. Keywords: Chaos and Dynamical Systems, Chemical Reaction Dynamics, Chemical Physics 1. Introduction In this paper we analyze the phase space structure and transport in a 2D caldera like potential energy surface (PES). The caldera PES is characterized by a flat region or shallow minimum at its center sur- rounded by potential walls and multiple symmetry related index one saddle points that allow entrance and exit from this intermediate region. This shape of the potential resembles the collapsed region within an 1 October 11, 2018 0:49 katsanikasandwiggins 2 Katsanikas and Wiggins errupted volcano (caldera), and this is the reason that Doering [Doering et al., 2002] and co-workers refer to this type of potential as a caldera. A caldera PES arises in many organic chemical reactions, such as the vinylcyclopropane-cyclopentene rearrangement [Baldwin, 2003; Goldschmidt et al., 1968], the stereomutation of cyclopropane [Doubleday et al., 1997], the degenerate rearrangement of bicyclo[3.1.0]hex-2-ene [Doubleday et al., 1999, 2006] or 5- methylenebicyclo[2.1.0]pentane [Reyes et al., 2002]. A key feature of this class of PES is that the stationary point corresponding to the intermediate often possesses higher symmetry than those of either the reactant or product stationary points. Under such circumstances, conventional statistical kinetic models would predict that the symmetry of the intermediate should be expressed in the product ratio. However, in the examples cited this is not observed, and it is the desire to understand this fact that has made analysis of the dynamics of reactions occuring on caldera potentials particularly important [Carpenter et al., 2016; Doering et al., 1968]. Fig. 1 shows the potential energy contours of a two dimensional caldera studied in Collins et al. [Collins et al., 2014]. In this model the symmetry-related 1-index saddles points provide a configuration space mechanism allowing for the the entrance of trajectories into and exit from the intermediate region of the caldera. Examining Fig. 1, one can easily suppose that the caldera is a decision point or crossroads in the reaction where selectivity between the four points of the saddles, that are indicated by black points, is determined. The subject of this paper is to investigate the phase space structure and transport associated mechanisms of the decision in the caldera and the origin of the selectivity between the four regions of the saddles. In order to address these questions Collins et al.[Collins et al., 2014] performed a detailed study of the trajectories in this two dimensional model. Broadly speaking, they identified two types of trajectory behaviour: dynamical matching [Collins et al., 2014] and trajectory trapping in the caldera. In this paper, we analyze the phase space mechanisms and the origin of the trajectory behaviour that was found in [Collins et al., 2014]. For this reason we study the phase space structure of the caldera and we use many methods from the theory of dynamical systems in order to have a deep insight of the dynamics of the system. We have computed all of the basic families of periodic orbits of the system and the invariant objects around them. We have found the mechanisms that are responsible for the entrance and the exit from the caldera and the phase space mechanisms for trapping and "dynamical matching". This could not have been achieved without the study of the phase space structure since these mechanisms are based on the geometrical objects in the phase space (invariant manifolds of the unstable periodic orbits). The structure of the paper is the following. In section 2 we describe the mathematical model for our system and the method of the surface of section that we used for the study of the phase space structure, and the families of the periodic orbits. In section 3 we describe our results for the four different cases of the structure of the phase space that we discovered in our study. Then we present the summary and our conclusions in section 4. For completeness we give an appendix A describing how we characterize the linear stability of periodic orbits and their bifurcations and an appendix B that describes the computation and characterization of periodic orbit dividing surfaces. 2. Model and Methods In this section we describe the 2D autonomous Hamiltonian system that is a model for the dynamics on the caldera. We introduce the method of Poincar´esurface of section used to investigate the phase space structure of our Hamiltonian System and we also present the families of periodic orbits of our system. 2.1. Hamiltonian system and equations of motion The Caldera potential has been used in a previous study from [Collins et al., 2014]. As we described in the 1, the topography of the potential is similar with the collapsed region of an erupted volcano. It has at the center a shallow minimum where there is a stable equilibrium point, the central minimum (see Table 2.1) . The shallow minimum of the potential is surrounded by potential walls. On these potential walls we have the existence of four 1-index saddles (two for lower values of Energy and other two for higher values of Energy, see Table 2.1) that allow the entrance into and exit from the intermediate region of the caldera. The four regions of the saddles are the only regions where molecules can enter into and exit from the caldera. The caldera has a stable equilibrium point at the center (that is represented by a black point October 11, 2018 0:49 katsanikasandwiggins Phase Space Structure and Transport in a Caldera Potential Energy Surface 3 Fig. 1. The stationary points (depicted by black points) and contours of the potential. at the center of the Fig. 1) and it is like a decision point in the reaction between four different paths that lead to four different regions of saddles and then to the exit from the caldera (that are represented by four black points in the Fig. 1). In addition, the entrance into the caldera is achieved through the four different regions of the saddles.The potential is: 2 4 V (x; y) = c1r + c2y − c3r cos[4θ] = 2 2 4 4 2 2 c1(y + x ) + c2y − c3(x + y − 6x y ) (1) 2 2 2 where r = x + y , cos[θ] = x=r.
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